Download Scheme for a coherently controlled pulsed electron gun F. Robicheaux

Robicheaux et al.
Vol. 15, No. 1 / January 1998 / J. Opt. Soc. Am. B
1
Scheme for a coherently controlled pulsed
electron gun
F. Robicheaux
Department of Physics, 206 Allison Laboratory, Auburn University Auburn, Alabama 36849-5311
G. M. Lankhuijzen and L. D. Noordam
FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands
Received September 15, 1997
When the outer electron of an alkali atom in an electric field is laser excited just above the field-induced ionization threshold, the electron ejection will not be instantaneous. Calculations show there are a number of
energy regions where, upon short-pulse laser excitation, the atom will eject a train of electron probability
pulses with the pulses being nearly equally spaced in time. This system can be the source of a picosecond
pulsed electron gun. We show that it is possible to coherently control the electron pulse frequency while the
pulse train is being emitted. © 1998 Optical Society of America [S0740-3224(98)03201-9]
OCIS codes: 030.1640, 320.5550, 140.3460.
1. INTRODUCTION
Recent work has focused on the study of the timedependent electron emission of highly excited alkali atoms in a strong electric field using experimental1–3 and
theoretical4,5 tools. The measurements on these Rydberg
atoms were performed by monitoring the time-dependent
flux of electrons that are naturally ejected from the atom.
The ejection occurs naturally because the electron has
enough energy to travel over the classical barrier to escape. However, because the energy is only slightly above
this barrier, the electron wave is trapped near the atom
for times of the order of picoseconds. Oscillatory structures in the ejected electron flux arise from the interference of the different energy components of the electron
wave.
In this paper we show that alkali atoms in an electric
field can be excited in such a way that a very long train
(.10) of electron pulses will be ejected with the pulses
having nearly uniform width and time separation. This
system may be interesting from a practical angle: it
could possibly be used as a source in a pulsed electron gun
(see Fig. 1). Just as a pulsed laser produces a series of
light pulses with a high repetition rate, the alkali atom in
a static field can produce coherently controlled electron
probability pulses at a high repetition rate. For the systems we have examined the pulse width can be 5 ps and
the time between pulses can be 15 ps. The period of the
pulses can be varied from 15 ps to 75 ps, allowing a large
tuneability in frequency. In contrast to short-pulse electron guns based on photoactivated metal cathodes6–9 or
scattering of an electron beam by a short optical pulse,10
the electron source discussed here will produce a series of
pulses rather than a single pulse. Moreover, the bursts
of probability within the pulse train are coherent and can
be manipulated while the emission is taking place. We
present calculations demonstrating that such a coherent
0740-3224/98/010001-05$10.00
train of electron pulses can indeed be produced by photoexcitation of autoionizing Rydberg states.
2. TIME STRUCTURE OF ATOMIC
ELECTRON EMISSION
The dynamics of autoionizing Rydberg states may be
fruitfully discussed from both a classical and a quantum
point of view. The classical hydrogen atom in a static
field evolves as follows: (1) An electron moves out from
the nucleus with an energy, E. (2) If the angle between
the electron’s initial velocity vector and the field direction
is smaller than a separatrix angle, which is the maximum
angle over which the electron can escape classically, the
electron will leave the region near the nucleus and travel
downfield. (3) If the initial angle is larger than the separatrix angle, the electron will be trapped forever in the region near the nucleus because this system separates in
parabolic coordinates. However, the presence of core
electrons in alkali atoms breaks the parabolic symmetry
near the nucleus. Each time the electron returns to
within a few Bohr radii of the nucleus, the orbit will precess, and the electron’s trajectory may now be at an angle
smaller than the separatrix angle. Thus it can leave the
atom. Therefore one might expect pulses of electrons,
with the time between the pulses related to the time required for the electron to return to the core region. The
highly excited electron exhibits two types of motion: (1) a
radial oscillation from the inner turning point to the outer
turning point, of the orbit, and (2) an oscillation of the angular momentum from its initial low l value all the way
up to l 5 n 2 1 and back. Core penetration, resulting in
scattering in the direction of the saddle point, is only possible whenever the angular momentum of the electron is
small [because the distance of closest approach is r min
; l(l 1 1)] and the electron is at the inner turning point
of its radial trajectory.
© 1998 Optical Society of America
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J. Opt. Soc. Am. B / Vol. 15, No. 1 / January 1998
Robicheaux et al.
tor with time spacings that depend on the inverse of the
energy spacing of the resonance states. The connection
between the classical and quantum description can be
made through the use of WKB and closed orbit theory.5
3. COMPUTATION OF THE ELECTRON
EMISSION SPECTRA
Fig. 1. Schematic representation of a coherent electron gun. A
short optical pulse excites a sample of gas-phase atoms in an
electric field to an autoionizing wave packet. The electron ejection after the laser excitation occurs in bursts. The time separation of the coherent electron wave packets is determined by the
angular-momentum oscillation time.
Fig. 2. Solid curve: the photoionization cross section of Na in a
static 1.2-kV/cm electric field with the laser polarized perpendicular to the static field direction. Dashed curve: spectrum of
the laser pulse.
The quantum description is very similar, yet interference effects can modify the emerging times in a qualitative manner.5 In the quantum description the wave
function for distances larger than a few Bohr radii is a superposition of functions in parabolic coordinates. The
coupling between the different channels arises in alkali
atoms through the phase shift of the radial function compared with the radial function for a hydrogen atom.11,12
This coupling allows states to decay more quickly above
the classical ionization threshold because it allows electron waves to escape the atom by tunneling and by scattering into channels that are classically open. The time
evolution of the electron wave is dominated by the spacings and widths of the resonances in the channels that
cannot decay classically. A detector downfield measures
an initial pulse of electron probability coming from the
part of the electron wave initially excited into open channels. Later pulses of electron probability enter the detec-
The computation techniques are along the lines of Refs. 4
and 5. The calculated time-dependent flux of electrons
may be obtained in a rather straightforward manner from
the time-dependent wave function. The wave packet is
generated through the action of a weak pulsed laser so
the time-dependent part of the wave function is calculated with first-order perturbation theory. The main difficulty is the calculation of the energy-dependent dipole
matrix elements. Another difficulty arises because the
usual superposition of energy eigenstates becomes an integration over an energy range since the states that are
being excited are in the continuum. However, the superposition of the infinite number of continuum states can be
performed with ordinary numerical integration techniques.
The experimental detection of the timedependent flux can be accomplished through the development and use of an atomic streak camera.13
In Fig. 2 we show the computed total photoionization
cross section as a function of energy for Na in a 1.2-kV/cm
static electric field. The Stark states may be classified by
the three quantum numbers n 1 (number of nodes in the
upfield parabolic coordinate), n 2 (number of nodes in the
downfield parabolic coordinate), and m (the angular momentum in the direction of the electric field); the principle
quantum number is n 5 n 1 1 n 2 1 u m u 1 1. These
states may be classified as 22,2,1 at E 5 25.715
3 1024 a.u., 18,8,1 at E 5 25.668 3 1024 a.u., 20,5,1 at
E 5 25.635 3 1024 a.u., and 17,10,1 at E 5 25.604
3 1024 a.u. Note that all of these states come from different n manifolds, which arises because this is the very
strong n-mixing regime.
The photon energy is measured relative to the zero field
threshold in Na. For this case the laser is polarized perpendicular to the static electric field. The fact that these
energy levels are nearly equally spaced means that when
they are excited by a ;15-ps laser pulse (that has a bandwidth comparable to the spacing of these levels), a wave
packet will be created that has very little dispersion.
The difference in the energy spacings is ;3% of the energy spacing. A wave packet constructed from these
states will not be bound to the atom forever but will be
ejected from the atom because these are autoionizing
states.
In Fig. 3 the flux of electrons ejected from the atom is
plotted as a function of time. It is clear from this figure
that the flux comes in pulses. The pulse period is given
by t 5 2 p /DE, where DE is the spacing of the resonances
shown in Fig. 2. The time between pulses for this case is
t ' 48 ps. The width of each electron pulse is roughly
t /N, where N is the number of states excited by the laser;
to make sharp electron pulses it is necessary to increase
N by finding several states with equal spacing. The
number of useful pulses that will emerge from this system
depends on how accurately the autoionizing resonances
Robicheaux et al.
Fig. 3. Electron flux as a function of time for excitation with the
laser spectrum shown in Fig. 2.
Fig. 4. Solid curve: the photoionization cross section of Na in a
static 1.6-kV/cm electric field with the laser polarized perpendicular to the static field direction. Dashed curve: spectrum of
the laser pulse.
Vol. 15, No. 1 / January 1998 / J. Opt. Soc. Am. B
3
pulses. If the resonances have a short lifetime, all of the
flux will leave the atom in a short time, and thus few
pulses will be ejected. However, the resonances should
not be too sharp because then there will be very little electron probability in each pulse. For example, these same
states in Li are much sharper than those in Fig. 2, giving
many more pulses but very little integrated flux in each
pulse.
In Fig. 4 a different energy range is presented. Again
the laser polarization is perpendicular to the static field,
but now the field strength equals 1.6 kV/cm and the optical pulse duration is ;8 ps. The states may be classified
as 19,3,1 at E 5 27.050 3 1024 a.u., 15,9,1 at E
5 26.995 3 1024 a.u.,
17,6,1
at
E 5 26.947
3 1024 a.u., and 14,11,1 at E 5 26.941 3 1024 a.u. In
Fig. 5 the ejected electron flux is plotted as a function of
time. Comparing Figs. 3 and 5, notice that the time scale
has changed by a factor of 2. The electron pulses are
separated by ;30 ps for this case. The difference in energy spacing for these states is ;2% of the energy spacing.
The calculations presented here result in electron
pulses of the order of 10 ps in duration. To obtain
shorter electron pulses, we can follow two strategies.
First, higher static fields (e.g., 10 kV/cm) will result in
somewhat shorter electron pulses (typically 1 ps). A second more dramatic improvement can be achieved by applying a two-step laser excitation scheme. The first laser
prepares the alkali atom in a stationary Stark state of a
low n manifold (e.g., n 5 6). The wave function of the
most blue Stark state (the highest energy state within the
n manifold) is strongly localized on the uphill side of the
potential. Photoionization by a second laser near threshold of such a blue Stark state is only possible to blue continuum states.14 Hence, if a short near-infrared laser
pulse of ;3 m m (e.g., Ref. 15) is used to photoionize the
n 5 6 blue Stark state, an autoionizing blue wave packet
will be excited. Since the energy spacing of blue states of
different n manifolds is large, the ejected electron pulses
will be short, well below 1 ps.
4. COHERENT CONTROL OF THE
EMISSION FREQUENCY
Fig. 5. Electron flux as a function of time for excitation with the
laser spectrum shown in Fig. 4.
are spaced and on the lifetimes of the autoionizing states.
If the resonances are not equally spaced, dispersion will
cause the wave packet to spread as it travels around the
atom; this will cause a smearing of the ejected electron
An appealing feature of such a train of electron pulses involves controlling the period of the pulses while the pulses
are being ejected. The cases we examined for more than
two resonances usually involved resonances from different n manifolds. This means that changing the field
strength will change the spacing in a complex manner
that will tend to cause the wave packet to disperse.
However, if the wave packet is constructed from two autoionizing states, the dispersion of the packet is not a
problem.
For two autoionizing states, the timedependent flux will have the simple form, exp(2Gt)(1
1 cos v t), if the two states have nearly the same autoionization width G and v 5 2 p /DE. Changing the
electric-field strength while the wave packet is on the
atom will cause the spacing to change, and thus v could
be controlled during the wave-packet experiment. As an
example of this, we discuss two states in Na that were
well separated from all other states over the field range of
4
J. Opt. Soc. Am. B / Vol. 15, No. 1 / January 1998
Robicheaux et al.
1.5–1.7 kV/cm. Over this small field range the separation of these two states changed by a factor of 3, which
implies a change of electron pulse period by a factor of 3.16
In Fig. 6 we present the results of a calculation that
shows the real-time tuning of the frequency of ejected
electron pulses. For this case we performed a direct numerical integration of Schrödinger’s equation using a discrete set of radial basis functions. The excitation of the
wave packet was treated as a Gaussian laser pulse that
excites states centered at energy, E 0 . The equation for
the wave-packet part of the pulse was then
F
i
G
]
2 H ~ t ! c ~ r, t ! 5 A ~ t ! x c g ~ r! ,
]t
(1)
where A(t) 5 exp@24 ln(2)t 2/t 2#, with t the FWHM of the
laser pulse, and the time-dependent Hamiltonian is given
by
H ~ t ! 5 H atom 2 E 0 1 F ~ t ! z.
(2)
H atom is the static zero-field atomic Hamiltonian, and F(t)
is a time-dependent electric field. For the case shown in
Fig. 6, E 0 5 20.00458747 a.u., t 5 8 ps, and F(t)
5 @ 2.917 3 1027 1 R(t)2.42 3 10215# a.u.
The function R(t) provides a slow ramp to the electric field. In
this calculation R(t) 5 0 for t , 30 ps. This means the
initial excitation is performed in a field of 1.5 kV/cm. After 30 ps, R(t) smoothly turns into a linear ramp using
the functional form R(t) 5 D 4 /(t 3c 1 D 3 ), where t c
5 30 ps 5 1.24 3 106 a.u. and D 5 t 2 t c , where both
D and t c are in atomic units. At large t, R(t) → t 2 t c ,
giving a linear ramp rate for the field of 0.514 V/(cm ps).
The two states that are excited are marked by arrows and
are the 23,2,1 state (at E 5 24.641 3 1024 a.u. at F
5 1.5 kV/cm) and the 22,4,1 state (at E 5 24.535
3 1024 a.u. at F 5 1.5 kV/cm); their energies at F
5 1.7 kV/cm are E 5 24.382 3 1024 a.u. and 24.353
3 1024 a.u.
The spacing of the electron probability pulses in Fig. 6
evolves from ;14 ps near 100 ps to ;40 ps near 400 ps.
This is a change in frequency of a factor of 2.5 with mod-
est ramp rates for the electric field. Thus one of the goals
of coherent control of atomic processes can be achieved in
this system: the experimental control of an atomic scale
process through the manipulation of a macroscopic variable. The control of the frequency of the electron probability pulses can be quite accurate since this is simply related to the energy spacing of the two levels.
5. APPLICATION FOR AN ELECTRON GUN
Such a train of electron pulses can, for instance, be used
for coherent hammering on surface phonons. We make a
coherent electron pulse train, and the repetition frequency of the electric pulses equals the phonon frequency.
By sending the train on a surface, the coupling to the
phonons will be large if there is a frequency match. Or in
other words, the incoming high-energy electron pulse
(e.g., 500 eV) will lose a marginal amount of energy to the
phonons (few megavolts). The energy loss is within the
energy spread of the short electron pulse. It is thus impossible to measure this loss with conventional electron
energy loss spectroscopy; however, the time separation of
a coherent train of pulses means that only a particular
energy can be effectively absorbed by the surface. This
technique of coherent hammering is similar to the train of
optical pulses that has been used for excitation of optical
phonons:17 the repetition rate of the optical pulse train
matches the phonon frequency, while the phonon energy
is within the energy bandwidth of an individual optical
pulse.
6. CONCLUSION
We have shown that laser-excited alkali atoms in a static
field can generate a series of electron pulses. The frequency of the pulses emitted can be controlled through
the field strength and the laser frequency that drives the
atom from its ground state into the wave-packet state.
Furthermore, we have shown that in certain special cases
the frequency of the electron pulses can be precisely manipulated while the electrons are being emitted. This allows a fine level of control of this system.
ACKNOWLEDGMENTS
F. Robicheaux was supported by the National Science
Foundation. G. Lankhuijzen and L. Noordam were supported by the Stichting voor Fundamenteel Onderzoek
der Materie, which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek.
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Fig. 6. Outgoing electron flux for a state initially in a 1.5-kV/cm
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laser pulse (dashed curve). The energy scale is in 1024 a.u.
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